Dynamical approach to Lévy processes

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This article discusses a dynamical approach to Lévy processes, which makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable Φy(t).

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8 p.

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Allegrini, Paolo; Grigolini, Paolo & West, Bruce J. November 1996.

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This article discusses a dynamical approach to Lévy processes, which makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable Φy(t).

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8 p.

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Copyright 1996 American Physical Society. The following article appeared in Physical Review E, 54:5, pp. 4760-4767, http://link.aps.org/doi/10.1103/PhysRevE.54.4760

Abstract: We derive the diffusion process generated by a correlated dichotomous fluctuating variable y starting from a Liouville-like equation by means of a projection procedure. This approach makes it possible to derive all statistical properties of the diffusion process from the correlation function of the dichotomous fluctuating variable Φy(t). Of special interest is that the distribution of the times of sojourn in the two states of the fluctuating process is proportional to d²Φy(t)/dt². Furthermore, in the special case where Φy(t) has an inverse power law, with the index β ranging from 0 to 1, thus making it nonintegrable, the authors show analytically that the statistics of the diffusing variable approximate in the long-time limit the α-stable Lévy distributions. The departure of the diffusion process of dynamical origin from the ideal condition of the Lévy statistics is established by means of a simple analytical expression. We note, first of all, that the characteristic function of a genuine Lévy process should be an exponential in time. We evaluate the correction to this exponential and show it to be expressed by a harmonic time oscillation modulated by the correlation function Φy(t). Since the characteristic function can be given a spectroscopic significance, we also discuss the relevance of the results within this context.

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  • Physical Review E, 1996, College Park: American Physical Society, pp. 4760

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  • Publication Title: Physical Review E
  • Volume: 54
  • Issue: 5
  • Page Start: 4760
  • Page End: 4767
  • Peer Reviewed: Yes

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  • November 1996

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  • Feb. 1, 2013, 9:58 a.m.

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  • July 18, 2013, 12:19 p.m.

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Allegrini, Paolo; Grigolini, Paolo & West, Bruce J. Dynamical approach to Lévy processes, article, November 1996; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc139498/: accessed November 21, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.